请以资深中国数学教师的身份，解析图片上习题的答案，要求讲解过程图文并茂，用不同的颜色标注，语言风格风趣幽默，简单易懂。---**Question 3:** x=1 是函数 $f(x) = \begin{cases} x-1, & x \le 1 \\ 3-x, & x > 1 \end{cases}$ 的 **Options:** A. 可去间断点 B. 跳跃间断点 C. 无穷间断点 D. 连续点

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Hello everyone! Today we're going to solve a classic problem about piecewise functions and discontinuities. We have a function defined in two pieces: x minus 1 for x less than or equal to 1, and 3 minus x for x greater than 1. We need to determine what type of discontinuity, if any, occurs at x equals 1. Let's visualize this function and analyze it step by step! Now let's check our three conditions for continuity step by step. First, we need to verify that the function is actually defined at x equals 1. Looking at our piecewise definition, when x equals 1, we use the first piece since 1 is less than or equal to 1. So f of 1 equals 1 minus 1, which equals 0. Great! The function is defined at x equals 1, and its value is 0. That's our first condition satisfied! Now for the crucial second condition: does the limit exist as x approaches 1? For this, we need to check both the left-hand and right-hand limits. From the left, we approach using the first piece x minus 1, giving us a limit of 0. From the right, we use the second piece 3 minus x, giving us a limit of 2. Since 0 does not equal 2, the left and right limits are different! This means the overall limit does not exist, and we have what's called a jump discontinuity! Now let's classify this discontinuity! We have four types to choose from. A removable discontinuity occurs when the limit exists but doesn't equal the function value. An infinite discontinuity happens when limits go to infinity. A continuous point satisfies all conditions. But our case has finite left and right limits that are different - 0 and 2. This is the classic definition of a jump discontinuity! The function literally jumps from one value to another at x equals 1. So our answer is B: Jump discontinuity! Perfect! Let's wrap up our analysis. We systematically checked all three conditions for continuity at x equals 1. The function is defined there with value 0, but the left and right limits are different - 0 and 2 respectively. This creates a jump discontinuity, where the function literally jumps from one value to another. So our final answer is B: Jump discontinuity. Great job working through this step by step! Remember, always check these conditions systematically when analyzing continuity.